Direct Variation
y changes directly in proportion to x.
Formula
Variables
Example
If y = 3x, then doubling x from 2 to 4 doubles y from 6 to 12.
How it works
Direct variation means one quantity stays in a fixed ratio with another, so every time x changes by some factor, y changes by the same factor. The constant k tells you how many units of y correspond to 1 unit of x, which is why the graph is a straight line through the origin when x = 0 gives y = 0.
When to use it
Use this formula when a situation has a constant rate, such as cost per item, distance at a constant speed, or pay per hour. It is also useful whenever one variable is exactly proportional to another and the relationship starts at zero.
Worked example, step by step
Suppose y varies directly with x and y = 18 when x = 6. Use y = kx and substitute the known values: 18 = k(6) Solve for k: k = 18 ÷ 6 = 3 So the formula is y = 3x Now find y when x = 10: y = 3(10) Compute: y = 30 Final answer: when x = 10, y = 30
Common mistakes to avoid
- Using direct variation when the relationship has a starting value other than 0
- Forgetting to find k from a known pair before trying to calculate new values
- Mixing up direct variation with inverse variation, where one variable decreases as the other increases
Did You Know?
Direct variation — double one, double the other — describes everything from Hooke's springs to fair pricing.
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